An explicit Skorokhod embedding for the age of Brownian excursions and Azema martingale

摘要

A general methodology allowing to solve the Skorokhod stopping problem for positive functionals of Brownian excursions, with the help of Brownian local time, is developed. The stopping times we consider have the following form: T-mu = inf{t > 0: F-t greater than or equal to phi(mu)(F)(L-t)}. As an application, the Skorokhod embedding problem for a number of functionals (F-t: t greater than or equal to 0), including the age (length) and the maximum (height) of excursions, is solved. Explicit formulae for the corresponding stopping times T-mu, such that F-Tmu similar to mu, are given. It is shown that the function phi(mu)(F) is the same for the maximum and for the age, phi(mu) = psi(mu)(-1), where psi(mu)(x) = integral([0,x])(y/(μ) over bar (y))dmu(y). The joint law of (g(Tmu), T-mu, L-Tmu), in the case of the age functional, is characterized. Examples for specific measures p are discussed. Finally, a randomized solution to the embedding problem for Azema martingale is deduced. Throughout the article, two possible approaches, using excursions and martingale theories, are presented in parallel. (C) 2003 Elsevier B.V. All rights reserved. [References: 27]